Über regulär-eindimensionale Räume

Nöbeling, Georg

doi:10.1007/bf01457922

Cited by 8 publications

(5 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…It is known that a regular space can be compactified to a regular space [11] (see also J. R. Isbell [Uniform spaces, Math. Surveys, no.…”

Section: Theorem a Planar Hereditarily Locally Connected Space Is Fmentioning

confidence: 99%

𝜎-connectedness in hereditarily locally connected spaces

Grispolakis¹,

Tymchatyn²

1979

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. B. Knaster, A. Lelek and J. Mycielski [Colloq. Math. 6 (1958), [227][228][229][230][231][232][233][234][235][236][237][238][239][240][241][242][243][244][245][246] had asked whether there exists a hereditarily locally connected planar set, which is the union of countably many disjoint arcs. They gave an example of a locally connected, connected planar set, which is the union of a countable sequence of disjoint arcs. Lelek proved in a paper in Fund. Math, in 1959, that connected subsets of planar hereditarily locally connected continua are weakly a-connected (i.e., they cannot be written as unions of countably many disjoint, closed connected subsets). In this paper we generalize the notion of finitely Susliman to noncompact spaces. We prove that there is a class of spaces, which includes the class of planar hereditarily locally connected spaces and the finitely Suslinian spaces, and which are weakly o-connected, thus, answering the above question in the negative. We also prove that arcwise connected, hereditarily locally connected, planar spaces are locally arcwise connected. This answers in the affirmative a question of Lelek [Colloq. Math. 36 (1976), 87-96].

show abstract

“…It is known that a regular space can be compactified to a regular space [11] (see also J. R. Isbell [Uniform spaces, Math. Surveys, no.…”

Section: Theorem a Planar Hereditarily Locally Connected Space Is Fmentioning

confidence: 99%

𝜎-connectedness in hereditarily locally connected spaces

Grispolakis¹,

Tymchatyn²

1979

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…So, for example, in [14] and [17] it is actually proved that there is no universal element in the family of all rational compacts. Earlier, in [16], it was proved that there is no universal element in the family of all peripherally finite compacts (the author uses another term), which is a subfamily of rational compacts.In and [9]. In these papers the following property of universal spaces is also considered.…”

mentioning

confidence: 97%

Some problems concerning universality in families ofn-Dimensional rational spaces

Iliadis¹

1996

J Math Sci

View full text Add to dashboard Cite

In this paper some problems concerning universality in families of n-dimensional rational spaces are given. Bibliography: 17 titles.All the spaces considered in this paper are metrically separable. A space T is called (isometrically) universal in a family Sp of spaces if T E Sp and for every X E Sp there exists an (isometric) embedding of X into T. Let Sp be a family of spaces. We denote by ~/(Sp) the family of spaces which is defined as follows: a space X belongs to ~(Sp) if X has a base B of open sets such that the boundary Bd(U) of any element U E B belongs to Sp.Suppose ~~ = Sp and ~n(Sp) = ~(~n-'(Sp)) for any n = 1,2, .... In what follows we denote by the family of all countable spaces (the empty set and finite sets are considered here as countable), The elements of the family ~n(M), n = 1, 2,..., are called spaces of rational dimension not greater than n (see, for example, [3] and [15]) or n-dimensional rational spaces. Obviously, ~I(~M) coincides with the family of rational spaces, which were considered by different authors under different names. So, for example, in [14] and [17] it is actually proved that there is no universal element in the family of all rational compacts. Earlier, in [16], it was proved that there is no universal element in the family of all peripherally finite compacts (the author uses another term), which is a subfamily of rational compacts.In and [9]. In these papers the following property of universal spaces is also considered. Let Sp be a family of spaces and (Sp)I its subfamily of power not greater than continuum. We say that a universal space 7' for the family Sp has the property of finite intersection with respect to the subfamily (Sp)I if for any X E Sp there exists an embedding ix : X ~ T such that the set iy(Y) f'3 iz(Z) is finite for any two different elements Y and Z of the family Sp, at least one of which belongs to (Sp)I.In [2], universality in the family of n-dimensional rational spaces for n = 1, 2,... is considered. The main result of [2] is the following.Theorem. /n the faxo51y ht/n(_/~/) there exists a universal element having the property of finite intersection with respect to a given subfamily of family ~:~'*( $VI), whose power is not greater than continuum.The aim of the present paper is to put certain questions concerning n-dimensional rational spaces, in particular, the questions connected with transfering results about rational spaces to this case.Let us begin with some definitions. Let a be a countable transfinite number. By 7(a) and m(a) we respectively denote a transfinite limit and a nonnegative integer such that a = 7(a) + re(a). Denote by z~/'(a) the family of all spaces M whose a-derivative (see [12], Vol. 1, w is empty. Let J~rc~ be the subfamily of JM(a) consisting of all compact elements of JM(a). For any nonnegative integer k, we let z~//ek(o~) (resp., ~/~(a)) denote the subfamily of ffv/(a) consisting of all elements M having a compact (resp., locally compact) extension with empty (~+k)-derivative. Obviously, J~/'(a) C fie/', ffffe~ C s176 and JMk...

show abstract

“…Το ερώτημα αυτό διατυ πωμένο για κανονικούς (γενικώτερα) χώρους απαντήθηκε αρνητικά από τον Nöbeling. (Βλέπε [44]). Στη συνέχεια ο Menger απέδειξε ότι και στην οικογένεια των κανονικών καμπυλών δεν υπάρχει καθολικό στοιχείο.…”

unclassified

“…Το αποτέλεσμα αυτό για α = 1 είναι το Θεώρημα Συμπαγοποίησης του Nöbeling σύμφωνα με το οκοίο για κάθε regular χώρο υπάρχει μία regular συμπαγοποίησή αυτού. (Βλέπε [44]).…”

unclassified

See 1 more Smart Citation

Προβληματα Καθολικοτητας Σε Οικογενειες Ρητων Χωρων

Φεγγοσ¹

View full text Add to dashboard Cite

Über regulär-eindimensionale Räume

Cited by 8 publications

References 2 publications

𝜎-connectedness in hereditarily locally connected spaces

𝜎-connectedness in hereditarily locally connected spaces

Some problems concerning universality in families ofn-Dimensional rational spaces

Προβληματα Καθολικοτητας Σε Οικογενειες Ρητων Χωρων

Contact Info

Product

Resources

About